Steem Riddle of the Week #7: Midpoints -- Solution Revealed + Prize Awarded

droopy (50)in #life • 6 years ago

Steem Riddle of the Week #7: Midpoints

For those who missed it, the full riddle is stated here: https://steemit.com/life/@droopy/steem-riddle-of-the-week-7-midpoints. Warning, spoilers are below! If you'd like to take a crack at the riddle on your own before seeing the solution, do not scroll down!

Solution

It turns out that it is actually impossible to find five points that meet the requirements of the riddle. A proof of this is below.

Remember that to find the mid points of two points (x1, y1) and (x2, y2), you can add their x and y components and divide by two. This means that the midpoint will have integer x and y coordinates if both x1+x2 is even and y1+y2 is even.

And of course the sum of two even numbers is even and the sum of two odd numbers is even.

This means that if x1 and x2 are both even or both odd and y1 and y2 are both even or both odd then the midpoint will have integer coordinates.

There are only 4 different possible arrangements of even/odd coordinates for x and y, they are:

(even, even)
(even, odd)
(odd, even)
(odd, odd)

We can place any of our 5 points into one of the categories above. If any two of our 5 points fall into the same category then their midpoint will have integer coordinates. Since there are 4 categories and 5 points there must be some category with two points in it. This is due to the pigeonhole principle.

Prize

This weeks one SBD prize went to @dpa-punch. Congratulations and thanks to everyone who participated!